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47
Eulerpoincaré equations and semidirect products with applications to continuum theories
 Adv. Math
, 1998
"... We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. ..."
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Cited by 125 (61 self)
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We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract KelvinNoether theorem for these equations. We also explore their relation with the theory of LiePoisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional CamassaHolm equations, which have many potentially interesting analytical properties. These
Discrete EulerPoincaré and LiePoisson equations
 Nonlinearity
, 1999
"... Abstract. In this paper, discrete analogues of EulerPoincaré and LiePoisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are Ginvariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the ..."
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Cited by 44 (5 self)
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Abstract. In this paper, discrete analogues of EulerPoincaré and LiePoisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are Ginvariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG,and a discrete Langragian L: G × G → R is constructed in such a way that the Ginvariance property is preserved. Reduction by G results in new “variational” principle for the reduced Lagrangian ℓ: G → R, and provides the discrete EulerPoincaré (DEP) equations. Reconstruction of these equations recovers the discrete EulerLagrange equations developed in [MPS 98, WM 97] which are naturally symplecticmomentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete LiePoisson (DLP) algorithm. It is shown that when G =SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser
The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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Cited by 38 (0 self)
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum
 Proceedings of the IEEE Conference on Control Applications
, 2005
"... Abstract — A numerical integrator is derived for a class of models that describe the attitude dynamics of a rigid body in the presence of an attitude dependent potential. The numerical integrator is obtained from a discrete variational principle, and exhibits excellent geometric conservation propert ..."
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Cited by 26 (19 self)
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Abstract — A numerical integrator is derived for a class of models that describe the attitude dynamics of a rigid body in the presence of an attitude dependent potential. The numerical integrator is obtained from a discrete variational principle, and exhibits excellent geometric conservation properties. In particular, by performing computations at the level of the Lie algebra, and updating the solution using the matrix exponential, the attitude automatically evolves on the rotation group embedded in the space of matrices. The geometric conservation properties of the numerical integrator imply long time numerical stability. We apply this variational integrator to the uncontrolled 3D pendulum, that is a rigid asymmetric body supported at a frictionless pivot acting under the influence of uniform gravity. Interesting dynamics of the 3D pendulum are exposed. I.
Lie group variational integrators for the full body problem
, 2007
"... We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motio ..."
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Cited by 21 (18 self)
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We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative frame is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.
A Variational Complex For Difference Equations
"... A variational complex for dierence equations is described that yields a characterization of dierence Euler Lagrange equations. The proof that the complex is locally exact is fully detailed. In particular, an analogue of the Poincare lemma for exact forms on a lattice is stated and proved, and homoto ..."
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Cited by 21 (1 self)
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A variational complex for dierence equations is described that yields a characterization of dierence Euler Lagrange equations. The proof that the complex is locally exact is fully detailed. In particular, an analogue of the Poincare lemma for exact forms on a lattice is stated and proved, and homotopy maps are given that allow one to calculate discrete Lagrangians for discrete EulerLagrange systems. Applications to the calculation of conservation laws and the continuum limit of our complex are outlined. 1 Contents 1
BÄCKLUND TRANSFORMATIONS FOR FINITEDIMENSIONAL INTEGRABLE SYSTEMS: A GEOMETRIC APPROACH
, 2000
"... We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parametrized by the points on the spectral curve(s) of the sys ..."
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Cited by 19 (3 self)
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We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parametrized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the AbelJacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multivalued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by adhoc methods, and we find Bäcklund transformations
Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top
 Comm. Math. Phys
, 1999
"... Abstract: We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a ..."
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Cited by 19 (1 self)
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Abstract: We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics – the Lagrange top. We recall the derivation of the Euler–Poinsot equations of motion both in the frame moving with the body and in the rest frame (the latter ones being less widely known). We find a discrete time Lagrange function turning into the known continuous time Lagrangian in the continuous limit, and elaborate both descriptions of the resulting discrete time system, namely in the body frame and in the rest frame. This system naturally inherits Poisson properties of the continuous time system, the integrals of motion being deformed. The discrete time Lax representations are also found. Kirchhoff’s kinetic analogy between elastic curves and motions of the Lagrange top is also generalised to the discrete context. 1.
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
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Cited by 17 (6 self)
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Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical
Symmetry Reduction of Discrete Lagrangian Mechanics on Lie groups
 J. GEOM. PHYS
, 2000
"... For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pullback of the LiePoisson structure on the dual of the Lie algebra � ∗ by the corresponding Legendre transform. The main result shown in this paper is that this structu ..."
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Cited by 15 (8 self)
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For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pullback of the LiePoisson structure on the dual of the Lie algebra � ∗ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2form ωL on G × G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced HamiltonJacobi equation are made. The rigid body is discussed as an example.